Rationalising the denominator with radicals

[hemmed]

Hi there,

I'd just like to check if my logic for rationalising this denominator containing a radical is correct. I have the following to solve:

$\displaystyle \large{\frac{x}{\sqrt[4]{x}\,}}$

Which I have then solved as follows (and this is listed as the correct answer in my book):
$\displaystyle \large{\frac{x}{\sqrt[4]{x}}\,\cdot\, \frac{\sqrt[4]{x^3}}{\sqrt[4]{x^3}}\,=\frac{x\sqrt[4]{x^3}}{\sqrt[4]{x^4}}\,=\frac{x\sqrt[4]{x^3}}{x}=\sqrt[4]{x^3}}\,$

I'd just like to check that my answer of $\displaystyle \large{{\sqrt[4]{x^3}}\,}$ is correct as the x's in the numerator and denominator have effectively cancelled themselves out, so you are therefore left with $\displaystyle \large{{\sqrt[4]{x^3}}\,}$ or in its full form... $\displaystyle \large{\frac{x\sqrt[4]{x^3}}{x}\,}$

 

[Harry_the_cat]

Hi there,

I'd just like to check if my logic for rationalising this denominator containing a radical is correct. I have the following to solve:

$\displaystyle \large{\frac{x}{\sqrt[4]{x}\,}}$

Which I have then solved as follows (and this is listed as the correct answer in my book):
$\displaystyle \large{\frac{x}{\sqrt[4]{x}}\,\cdot\, \frac{\sqrt[4]{x^3}}{\sqrt[4]{x^3}}\,=\frac{x\sqrt[4]{x^3}}{\sqrt[4]{x^4}}\,=\frac{x\sqrt[4]{x^3}}{x}=\sqrt[4]{x^3}}\,$

I'd just like to check that my answer of $\displaystyle \large{{\sqrt[4]{x^3}}\,}$ is correct as the x's in the numerator and denominator have effectively cancelled themselves out, so you are therefore left with $\displaystyle \large{{\sqrt[4]{x^3}}\,}$ or in its full form... $\displaystyle \large{\frac{x\sqrt[4]{x^3}}{x}\,}$

Yes that's correct.
Another way to do it is to use exponents:
$\displaystyle \frac{x}{x^{(1/4)}} = x* x^{({-1/4})} = x^{({1 - 1/4})} = x^{({3/4})}$
which is what you have in radical form.

 

[hemmed]

Yes that's correct.
Another way to do it is to use exponents:
$\displaystyle \frac{x}{x^{(1/4)}} = x* x^{({-1/4})} = x^{({1 - 1/4})} = x^{({3/4})}$
which is what you have in radical form.

Excellent - thank you!